2isqrt (example 3.6)

Percentage Accurate: 38.4% → 99.2%

Time: 17.5s

Alternatives: 12

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 38.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{2}} + \frac{0.0390625}{{x}^{4}}\right)\right)\right) \cdot {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   (/ 0.0625 (pow x 3.0))
   (- (/ 0.5 x) (+ (/ 0.125 (pow x 2.0)) (/ 0.0390625 (pow x 4.0)))))
  (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return ((0.0625 / pow(x, 3.0)) + ((0.5 / x) - ((0.125 / pow(x, 2.0)) + (0.0390625 / pow(x, 4.0))))) * pow((x + 1.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.0625d0 / (x ** 3.0d0)) + ((0.5d0 / x) - ((0.125d0 / (x ** 2.0d0)) + (0.0390625d0 / (x ** 4.0d0))))) * ((x + 1.0d0) ** (-0.5d0))
end function

Java

public static double code(double x) {
	return ((0.0625 / Math.pow(x, 3.0)) + ((0.5 / x) - ((0.125 / Math.pow(x, 2.0)) + (0.0390625 / Math.pow(x, 4.0))))) * Math.pow((x + 1.0), -0.5);
}

Python

def code(x):
	return ((0.0625 / math.pow(x, 3.0)) + ((0.5 / x) - ((0.125 / math.pow(x, 2.0)) + (0.0390625 / math.pow(x, 4.0))))) * math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64(Float64(Float64(0.0625 / (x ^ 3.0)) + Float64(Float64(0.5 / x) - Float64(Float64(0.125 / (x ^ 2.0)) + Float64(0.0390625 / (x ^ 4.0))))) * (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = ((0.0625 / (x ^ 3.0)) + ((0.5 / x) - ((0.125 / (x ^ 2.0)) + (0.0390625 / (x ^ 4.0))))) * ((x + 1.0) ^ -0.5);
end

Wolfram

code[x_] := N[(N[(N[(0.0625 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / x), $MachinePrecision] - N[(N[(0.125 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.0390625 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\left(\left(0.0625 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. associate--l+ 98.9%

      \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{1}{{x}^{3}} + \left(0.5 \cdot \frac{1}{x} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   2. associate-*r/ 98.9%

      \[\leadsto \left(\color{blue}{\frac{0.0625 \cdot 1}{{x}^{3}}} + \left(0.5 \cdot \frac{1}{x} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   3. metadata-eval 98.9%

      \[\leadsto \left(\frac{\color{blue}{0.0625}}{{x}^{3}} + \left(0.5 \cdot \frac{1}{x} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   4. associate-*r/ 98.9%

      \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. metadata-eval 98.9%

      \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{\color{blue}{0.5}}{x} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. +-commutative 98.9%

      \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{4}}\right)}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. associate-*r/ 98.9%

      \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}} + 0.0390625 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   8. metadata-eval 98.9%

      \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{\color{blue}{0.125}}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   9. associate-*r/ 98.9%

      \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{2}} + \color{blue}{\frac{0.0390625 \cdot 1}{{x}^{4}}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   10. metadata-eval 98.9%

       \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{2}} + \frac{\color{blue}{0.0390625}}{{x}^{4}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

9. Simplified 98.9%

   \[\leadsto \color{blue}{\left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{2}} + \frac{0.0390625}{{x}^{4}}\right)\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

10. Final simplification 98.9%

    \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{2}} + \frac{0.0390625}{{x}^{4}}\right)\right)\right) \cdot {\left(x + 1\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{-0.5} \cdot \left(\left(\frac{0.0625}{{x}^{3}} + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (+ x 1.0) -0.5)
  (- (+ (/ 0.0625 (pow x 3.0)) (/ 0.5 x)) (/ 0.125 (pow x 2.0)))))

C

double code(double x) {
	return pow((x + 1.0), -0.5) * (((0.0625 / pow(x, 3.0)) + (0.5 / x)) - (0.125 / pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) ** (-0.5d0)) * (((0.0625d0 / (x ** 3.0d0)) + (0.5d0 / x)) - (0.125d0 / (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.pow((x + 1.0), -0.5) * (((0.0625 / Math.pow(x, 3.0)) + (0.5 / x)) - (0.125 / Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.pow((x + 1.0), -0.5) * (((0.0625 / math.pow(x, 3.0)) + (0.5 / x)) - (0.125 / math.pow(x, 2.0)))

Julia

function code(x)
	return Float64((Float64(x + 1.0) ^ -0.5) * Float64(Float64(Float64(0.0625 / (x ^ 3.0)) + Float64(0.5 / x)) - Float64(0.125 / (x ^ 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) * (((0.0625 / (x ^ 3.0)) + (0.5 / x)) - (0.125 / (x ^ 2.0)));
end

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(N[(0.0625 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] - N[(0.125 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 98.7%

   \[\leadsto \color{blue}{\left(\left(0.0625 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. +-commutative 98.7%

      \[\leadsto \left(\color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{3}}\right)} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   2. associate-*r/ 98.7%

      \[\leadsto \left(\left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.0625 \cdot \frac{1}{{x}^{3}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   3. metadata-eval 98.7%

      \[\leadsto \left(\left(\frac{\color{blue}{0.5}}{x} + 0.0625 \cdot \frac{1}{{x}^{3}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   4. associate-*r/ 98.7%

      \[\leadsto \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{3}}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. metadata-eval 98.7%

      \[\leadsto \left(\left(\frac{0.5}{x} + \frac{\color{blue}{0.0625}}{{x}^{3}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. associate-*r/ 98.7%

      \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.0625}{{x}^{3}}\right) - \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 98.7%

      \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.0625}{{x}^{3}}\right) - \frac{\color{blue}{0.125}}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

9. Simplified 98.7%

   \[\leadsto \color{blue}{\left(\left(\frac{0.5}{x} + \frac{0.0625}{{x}^{3}}\right) - \frac{0.125}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

10. Final simplification 98.7%

    \[\leadsto {\left(x + 1\right)}^{-0.5} \cdot \left(\left(\frac{0.0625}{{x}^{3}} + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{2}}\right) \]
11. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot {x}^{-2} + \frac{-0.5}{x}}{\sqrt{x + 1}}}{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ (* 0.125 (pow x -2.0)) (/ -0.5 x)) (sqrt (+ x 1.0))) -1.0))

C

double code(double x) {
	return (((0.125 * pow(x, -2.0)) + (-0.5 / x)) / sqrt((x + 1.0))) / -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((0.125d0 * (x ** (-2.0d0))) + ((-0.5d0) / x)) / sqrt((x + 1.0d0))) / (-1.0d0)
end function

Java

public static double code(double x) {
	return (((0.125 * Math.pow(x, -2.0)) + (-0.5 / x)) / Math.sqrt((x + 1.0))) / -1.0;
}

Python

def code(x):
	return (((0.125 * math.pow(x, -2.0)) + (-0.5 / x)) / math.sqrt((x + 1.0))) / -1.0

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.125 * (x ^ -2.0)) + Float64(-0.5 / x)) / sqrt(Float64(x + 1.0))) / -1.0)
end

MATLAB

function tmp = code(x)
	tmp = (((0.125 * (x ^ -2.0)) + (-0.5 / x)) / sqrt((x + 1.0))) / -1.0;
end

Wolfram

code[x_] := N[(N[(N[(N[(0.125 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 98.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.125 \cdot \frac{1}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. associate-*r/ 98.3%

      \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   2. metadata-eval 98.3%

      \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   3. associate-*r/ 98.3%

      \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   4. metadata-eval 98.3%

      \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.125}}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

9. Simplified 98.3%

   \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
10. Step-by-step derivation

    1. /-rgt-identity 98.3%

       \[\leadsto \color{blue}{\frac{\left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}}{1}} \]

    2. frac-2neg 98.3%

       \[\leadsto \color{blue}{\frac{-\left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}}{-1}} \]

11. Applied egg-rr 98.3%

    \[\leadsto \color{blue}{\frac{-\frac{\frac{0.5}{x} + -0.125 \cdot {x}^{-2}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}{-1}} \]
12. Step-by-step derivation

    1. distribute-neg-frac 98.3%

       \[\leadsto \frac{\color{blue}{\frac{-\left(\frac{0.5}{x} + -0.125 \cdot {x}^{-2}\right)}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}}{-1} \]

    2. +-commutative 98.3%

       \[\leadsto \frac{\frac{-\color{blue}{\left(-0.125 \cdot {x}^{-2} + \frac{0.5}{x}\right)}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}{-1} \]

    3. distribute-neg-in 98.3%

       \[\leadsto \frac{\frac{\color{blue}{\left(--0.125 \cdot {x}^{-2}\right) + \left(-\frac{0.5}{x}\right)}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}{-1} \]

    4. distribute-lft-neg-in 98.3%

       \[\leadsto \frac{\frac{\color{blue}{\left(--0.125\right) \cdot {x}^{-2}} + \left(-\frac{0.5}{x}\right)}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}{-1} \]

    5. metadata-eval 98.3%

       \[\leadsto \frac{\frac{\color{blue}{0.125} \cdot {x}^{-2} + \left(-\frac{0.5}{x}\right)}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}{-1} \]

    6. distribute-neg-frac 98.3%

       \[\leadsto \frac{\frac{0.125 \cdot {x}^{-2} + \color{blue}{\frac{-0.5}{x}}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}{-1} \]

    7. metadata-eval 98.3%

       \[\leadsto \frac{\frac{0.125 \cdot {x}^{-2} + \frac{\color{blue}{-0.5}}{x}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}{-1} \]

    8. hypot-undefine 98.2%

       \[\leadsto \frac{\frac{0.125 \cdot {x}^{-2} + \frac{-0.5}{x}}{\color{blue}{\sqrt{1 \cdot 1 + \sqrt{x} \cdot \sqrt{x}}}}}{-1} \]

    9. metadata-eval 98.2%

       \[\leadsto \frac{\frac{0.125 \cdot {x}^{-2} + \frac{-0.5}{x}}{\sqrt{\color{blue}{1} + \sqrt{x} \cdot \sqrt{x}}}}{-1} \]

    10. rem-square-sqrt 98.2%

        \[\leadsto \frac{\frac{0.125 \cdot {x}^{-2} + \frac{-0.5}{x}}{\sqrt{1 + \color{blue}{x}}}}{-1} \]

    11. +-commutative 98.2%

        \[\leadsto \frac{\frac{0.125 \cdot {x}^{-2} + \frac{-0.5}{x}}{\sqrt{\color{blue}{x + 1}}}}{-1} \]

13. Simplified 98.2%

    \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot {x}^{-2} + \frac{-0.5}{x}}{\sqrt{x + 1}}}{-1}} \]

14. Final simplification 98.2%

    \[\leadsto \frac{\frac{0.125 \cdot {x}^{-2} + \frac{-0.5}{x}}{\sqrt{x + 1}}}{-1} \]
15. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (+ x 1.0) -0.5) (- (/ 0.5 x) (/ 0.125 (pow x 2.0)))))

C

double code(double x) {
	return pow((x + 1.0), -0.5) * ((0.5 / x) - (0.125 / pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) ** (-0.5d0)) * ((0.5d0 / x) - (0.125d0 / (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.pow((x + 1.0), -0.5) * ((0.5 / x) - (0.125 / Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.pow((x + 1.0), -0.5) * ((0.5 / x) - (0.125 / math.pow(x, 2.0)))

Julia

function code(x)
	return Float64((Float64(x + 1.0) ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.125 / (x ^ 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) * ((0.5 / x) - (0.125 / (x ^ 2.0)));
end

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.125 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 98.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.125 \cdot \frac{1}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. associate-*r/ 98.3%

      \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   2. metadata-eval 98.3%

      \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   3. associate-*r/ 98.3%

      \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   4. metadata-eval 98.3%

      \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.125}}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

9. Simplified 98.3%

   \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

10. Final simplification 98.3%

    \[\leadsto {\left(x + 1\right)}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.125}{{x}^{2}}\right) \]
11. Add Preprocessing

Alternative 5: 37.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.0390625}{{x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.5e+153) (/ 0.5 x) (/ -0.0390625 (pow x 4.0))))

C

double code(double x) {
	double tmp;
	if (x <= 6.5e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = -0.0390625 / pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.5d+153) then
        tmp = 0.5d0 / x
    else
        tmp = (-0.0390625d0) / (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.5e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = -0.0390625 / Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.5e+153:
		tmp = 0.5 / x
	else:
		tmp = -0.0390625 / math.pow(x, 4.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.5e+153)
		tmp = Float64(0.5 / x);
	else
		tmp = Float64(-0.0390625 / (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.5e+153)
		tmp = 0.5 / x;
	else
		tmp = -0.0390625 / (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.5e+153], N[(0.5 / x), $MachinePrecision], N[(-0.0390625 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.49999999999999972e153

   1. Initial program 11.8%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-sub 11.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

      2. div-inv 11.8%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

      3. *-un-lft-identity 11.8%

         \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

      4. +-commutative 11.8%

         \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

      5. *-rgt-identity 11.8%

         \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

      6. metadata-eval 11.8%

         \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

      7. frac-times 11.8%

         \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

      8. associate-*l/ 11.8%

         \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

      9. inv-pow 11.8%

         \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

      10. *-un-lft-identity 11.8%

          \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

      11. sqrt-pow2 11.8%

          \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

      12. +-commutative 11.8%

          \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

      13. metadata-eval 11.8%

          \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

   4. Applied egg-rr 11.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 11.8%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

      2. *-rgt-identity 11.8%

         \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

      3. times-frac 11.8%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

      4. div-sub 11.8%

         \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      5. sub-neg 11.8%

         \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      6. *-inverses 11.8%

         \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      7. metadata-eval 11.8%

         \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. Simplified 11.8%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   7. Taylor expanded in x around inf 94.5%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   8. Taylor expanded in x around 0 8.7%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   if 6.49999999999999972e153 < x

   1. Initial program 71.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-sub 71.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

      2. div-inv 71.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

      3. *-un-lft-identity 71.0%

         \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

      4. +-commutative 71.0%

         \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

      5. *-rgt-identity 71.0%

         \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

      6. metadata-eval 71.0%

         \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

      7. frac-times 71.0%

         \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

      8. associate-*l/ 71.0%

         \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

      9. inv-pow 71.0%

         \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

      10. *-un-lft-identity 71.0%

          \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

      11. sqrt-pow2 71.0%

          \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

      12. +-commutative 71.0%

          \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

      13. metadata-eval 71.0%

          \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

   4. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 71.0%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

      2. *-rgt-identity 71.0%

         \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

      3. times-frac 71.0%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

      4. div-sub 71.0%

         \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      5. sub-neg 71.0%

         \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      6. *-inverses 71.0%

         \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      7. metadata-eval 71.0%

         \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. Simplified 71.0%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   7. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\left(\left(0.0625 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
   8. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{\left(0.0625 \cdot \frac{1}{{x}^{3}} + \left(0.5 \cdot \frac{1}{x} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      2. associate-*r/ 99.9%

         \[\leadsto \left(\color{blue}{\frac{0.0625 \cdot 1}{{x}^{3}}} + \left(0.5 \cdot \frac{1}{x} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      3. metadata-eval 99.9%

         \[\leadsto \left(\frac{\color{blue}{0.0625}}{{x}^{3}} + \left(0.5 \cdot \frac{1}{x} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      4. associate-*r/ 99.9%

         \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      5. metadata-eval 99.9%

         \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{\color{blue}{0.5}}{x} - \left(0.0390625 \cdot \frac{1}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      6. +-commutative 99.9%

         \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{4}}\right)}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      7. associate-*r/ 99.9%

         \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}} + 0.0390625 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      8. metadata-eval 99.9%

         \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{\color{blue}{0.125}}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      9. associate-*r/ 99.9%

         \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{2}} + \color{blue}{\frac{0.0390625 \cdot 1}{{x}^{4}}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

      10. metadata-eval 99.9%

          \[\leadsto \left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{2}} + \frac{\color{blue}{0.0390625}}{{x}^{4}}\right)\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   9. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\frac{0.0625}{{x}^{3}} + \left(\frac{0.5}{x} - \left(\frac{0.125}{{x}^{2}} + \frac{0.0390625}{{x}^{4}}\right)\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   10. Taylor expanded in x around 0 71.0%

       \[\leadsto \color{blue}{\frac{-0.0390625}{{x}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.0390625}{{x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x} \cdot {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ 0.5 x) (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return (0.5 / x) * pow((x + 1.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 / x) * ((x + 1.0d0) ** (-0.5d0))
end function

Java

public static double code(double x) {
	return (0.5 / x) * Math.pow((x + 1.0), -0.5);
}

Python

def code(x):
	return (0.5 / x) * math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64(Float64(0.5 / x) * (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 / x) * ((x + 1.0) ^ -0.5);
end

Wolfram

code[x_] := N[(N[(0.5 / x), $MachinePrecision] * N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 97.1%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. frac-2neg 97.1%

      \[\leadsto \color{blue}{\frac{-0.5}{-x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   2. metadata-eval 97.1%

      \[\leadsto \frac{\color{blue}{-0.5}}{-x} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   3. frac-times 97.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot {\left(1 + x\right)}^{-0.5}}{\left(-x\right) \cdot 1}} \]

   4. +-commutative 97.2%

      \[\leadsto \frac{-0.5 \cdot {\color{blue}{\left(x + 1\right)}}^{-0.5}}{\left(-x\right) \cdot 1} \]

9. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot {\left(x + 1\right)}^{-0.5}}{\left(-x\right) \cdot 1}} \]
10. Step-by-step derivation

    1. times-frac 97.1%

       \[\leadsto \color{blue}{\frac{-0.5}{-x} \cdot \frac{{\left(x + 1\right)}^{-0.5}}{1}} \]

    2. neg-mul-1 97.1%

       \[\leadsto \frac{-0.5}{\color{blue}{-1 \cdot x}} \cdot \frac{{\left(x + 1\right)}^{-0.5}}{1} \]

    3. associate-/r* 97.1%

       \[\leadsto \color{blue}{\frac{\frac{-0.5}{-1}}{x}} \cdot \frac{{\left(x + 1\right)}^{-0.5}}{1} \]

    4. metadata-eval 97.1%

       \[\leadsto \frac{\color{blue}{0.5}}{x} \cdot \frac{{\left(x + 1\right)}^{-0.5}}{1} \]

    5. /-rgt-identity 97.1%

       \[\leadsto \frac{0.5}{x} \cdot \color{blue}{{\left(x + 1\right)}^{-0.5}} \]

11. Simplified 97.1%

    \[\leadsto \color{blue}{\frac{0.5}{x} \cdot {\left(x + 1\right)}^{-0.5}} \]

12. Final simplification 97.1%

    \[\leadsto \frac{0.5}{x} \cdot {\left(x + 1\right)}^{-0.5} \]
13. Add Preprocessing

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(x + 1\right)}^{-0.5}}{x \cdot 2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow (+ x 1.0) -0.5) (* x 2.0)))

C

double code(double x) {
	return pow((x + 1.0), -0.5) / (x * 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) ** (-0.5d0)) / (x * 2.0d0)
end function

Java

public static double code(double x) {
	return Math.pow((x + 1.0), -0.5) / (x * 2.0);
}

Python

def code(x):
	return math.pow((x + 1.0), -0.5) / (x * 2.0)

Julia

function code(x)
	return Float64((Float64(x + 1.0) ^ -0.5) / Float64(x * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) / (x * 2.0);
end

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 97.1%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. /-rgt-identity 97.1%

      \[\leadsto \frac{0.5}{x} \cdot \color{blue}{{\left(1 + x\right)}^{-0.5}} \]

   2. *-commutative 97.1%

      \[\leadsto \color{blue}{{\left(1 + x\right)}^{-0.5} \cdot \frac{0.5}{x}} \]

   3. clear-num 97.1%

      \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \color{blue}{\frac{1}{\frac{x}{0.5}}} \]

   4. un-div-inv 97.2%

      \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{-0.5}}{\frac{x}{0.5}}} \]

   5. +-commutative 97.2%

      \[\leadsto \frac{{\color{blue}{\left(x + 1\right)}}^{-0.5}}{\frac{x}{0.5}} \]

   6. div-inv 97.2%

      \[\leadsto \frac{{\left(x + 1\right)}^{-0.5}}{\color{blue}{x \cdot \frac{1}{0.5}}} \]

   7. metadata-eval 97.2%

      \[\leadsto \frac{{\left(x + 1\right)}^{-0.5}}{x \cdot \color{blue}{2}} \]

9. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{-0.5}}{x \cdot 2}} \]

10. Final simplification 97.2%

    \[\leadsto \frac{{\left(x + 1\right)}^{-0.5}}{x \cdot 2} \]
11. Add Preprocessing

Alternative 8: 96.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (* x (sqrt (+ x 1.0)))))

C

double code(double x) {
	return 0.5 / (x * sqrt((x + 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 / (x * sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return 0.5 / (x * Math.sqrt((x + 1.0)));
}

Python

def code(x):
	return 0.5 / (x * math.sqrt((x + 1.0)))

Julia

function code(x)
	return Float64(0.5 / Float64(x * sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / (x * sqrt((x + 1.0)));
end

Wolfram

code[x_] := N[(0.5 / N[(x * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 97.1%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. frac-2neg 97.1%

      \[\leadsto \color{blue}{\frac{-0.5}{-x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   2. metadata-eval 97.1%

      \[\leadsto \frac{\color{blue}{-0.5}}{-x} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   3. frac-times 97.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot {\left(1 + x\right)}^{-0.5}}{\left(-x\right) \cdot 1}} \]

   4. +-commutative 97.2%

      \[\leadsto \frac{-0.5 \cdot {\color{blue}{\left(x + 1\right)}}^{-0.5}}{\left(-x\right) \cdot 1} \]

9. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot {\left(x + 1\right)}^{-0.5}}{\left(-x\right) \cdot 1}} \]
10. Step-by-step derivation

    1. *-rgt-identity 97.2%

       \[\leadsto \frac{-0.5 \cdot {\left(x + 1\right)}^{-0.5}}{\color{blue}{-x}} \]

    2. associate-/l* 96.9%

       \[\leadsto \color{blue}{\frac{-0.5}{\frac{-x}{{\left(x + 1\right)}^{-0.5}}}} \]

11. Simplified 96.9%

    \[\leadsto \color{blue}{\frac{-0.5}{\frac{-x}{{\left(x + 1\right)}^{-0.5}}}} \]
12. Step-by-step derivation

    1. *-un-lft-identity 96.9%

       \[\leadsto \color{blue}{1 \cdot \frac{-0.5}{\frac{-x}{{\left(x + 1\right)}^{-0.5}}}} \]

    2. associate-/r/ 97.1%

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{-0.5}{-x} \cdot {\left(x + 1\right)}^{-0.5}\right)} \]

    3. metadata-eval 97.1%

       \[\leadsto 1 \cdot \left(\frac{\color{blue}{-0.5}}{-x} \cdot {\left(x + 1\right)}^{-0.5}\right) \]

    4. frac-2neg 97.1%

       \[\leadsto 1 \cdot \left(\color{blue}{\frac{0.5}{x}} \cdot {\left(x + 1\right)}^{-0.5}\right) \]

    5. metadata-eval 97.1%

       \[\leadsto 1 \cdot \left(\frac{0.5}{x} \cdot {\left(x + 1\right)}^{\color{blue}{\left(-0.5\right)}}\right) \]

    6. pow-flip 97.0%

       \[\leadsto 1 \cdot \left(\frac{0.5}{x} \cdot \color{blue}{\frac{1}{{\left(x + 1\right)}^{0.5}}}\right) \]

    7. pow1/2 97.0%

       \[\leadsto 1 \cdot \left(\frac{0.5}{x} \cdot \frac{1}{\color{blue}{\sqrt{x + 1}}}\right) \]

    8. un-div-inv 97.0%

       \[\leadsto 1 \cdot \color{blue}{\frac{\frac{0.5}{x}}{\sqrt{x + 1}}} \]

13. Applied egg-rr 97.0%

    \[\leadsto \color{blue}{1 \cdot \frac{\frac{0.5}{x}}{\sqrt{x + 1}}} \]
14. Step-by-step derivation

    1. *-lft-identity 97.0%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{x}}{\sqrt{x + 1}}} \]

    2. associate-/r* 96.8%

       \[\leadsto \color{blue}{\frac{0.5}{x \cdot \sqrt{x + 1}}} \]

15. Simplified 96.8%

    \[\leadsto \color{blue}{\frac{0.5}{x \cdot \sqrt{x + 1}}} \]

16. Final simplification 96.8%

    \[\leadsto \frac{0.5}{x \cdot \sqrt{x + 1}} \]
17. Add Preprocessing

Alternative 9: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 x) (sqrt (+ x 1.0))))

C

double code(double x) {
	return (0.5 / x) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 / x) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return (0.5 / x) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return (0.5 / x) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(0.5 / x) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 / x) / sqrt((x + 1.0));
end

Wolfram

code[x_] := N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 97.1%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. clear-num 97.0%

      \[\leadsto \frac{0.5}{x} \cdot \color{blue}{\frac{1}{\frac{1}{{\left(1 + x\right)}^{-0.5}}}} \]

   2. pow-flip 97.0%

      \[\leadsto \frac{0.5}{x} \cdot \frac{1}{\color{blue}{{\left(1 + x\right)}^{\left(--0.5\right)}}} \]

   3. metadata-eval 97.0%

      \[\leadsto \frac{0.5}{x} \cdot \frac{1}{{\left(1 + x\right)}^{\color{blue}{0.5}}} \]

   4. pow1/2 97.0%

      \[\leadsto \frac{0.5}{x} \cdot \frac{1}{\color{blue}{\sqrt{1 + x}}} \]

   5. frac-times 96.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{x \cdot \sqrt{1 + x}}} \]

   6. metadata-eval 96.8%

      \[\leadsto \frac{\color{blue}{0.5}}{x \cdot \sqrt{1 + x}} \]

   7. add-sqr-sqrt 96.8%

      \[\leadsto \frac{0.5}{x \cdot \sqrt{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

   8. hypot-1-def 96.8%

      \[\leadsto \frac{0.5}{x \cdot \color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]

9. Applied egg-rr 96.8%

   \[\leadsto \color{blue}{\frac{0.5}{x \cdot \mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
10. Step-by-step derivation

    1. associate-/r* 97.0%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{x}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]

    2. hypot-undefine 97.0%

       \[\leadsto \frac{\frac{0.5}{x}}{\color{blue}{\sqrt{1 \cdot 1 + \sqrt{x} \cdot \sqrt{x}}}} \]

    3. metadata-eval 97.0%

       \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\color{blue}{1} + \sqrt{x} \cdot \sqrt{x}}} \]

    4. rem-square-sqrt 97.0%

       \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{1 + \color{blue}{x}}} \]

    5. +-commutative 97.0%

       \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\color{blue}{x + 1}}} \]

11. Simplified 97.0%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{x}}{\sqrt{x + 1}}} \]

12. Final simplification 97.0%

    \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{x + 1}} \]
13. Add Preprocessing

Alternative 10: 36.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.0625}{{x}^{3}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.0625 (pow x 3.0)))

C

double code(double x) {
	return 0.0625 / pow(x, 3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0625d0 / (x ** 3.0d0)
end function

Java

public static double code(double x) {
	return 0.0625 / Math.pow(x, 3.0);
}

Python

def code(x):
	return 0.0625 / math.pow(x, 3.0)

Julia

function code(x)
	return Float64(0.0625 / (x ^ 3.0))
end

MATLAB

function tmp = code(x)
	tmp = 0.0625 / (x ^ 3.0);
end

Wolfram

code[x_] := N[(0.0625 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 98.7%

   \[\leadsto \color{blue}{\left(\left(0.0625 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]
8. Step-by-step derivation

   1. +-commutative 98.7%

      \[\leadsto \left(\color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{3}}\right)} - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   2. associate-*r/ 98.7%

      \[\leadsto \left(\left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.0625 \cdot \frac{1}{{x}^{3}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   3. metadata-eval 98.7%

      \[\leadsto \left(\left(\frac{\color{blue}{0.5}}{x} + 0.0625 \cdot \frac{1}{{x}^{3}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   4. associate-*r/ 98.7%

      \[\leadsto \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{3}}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. metadata-eval 98.7%

      \[\leadsto \left(\left(\frac{0.5}{x} + \frac{\color{blue}{0.0625}}{{x}^{3}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. associate-*r/ 98.7%

      \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.0625}{{x}^{3}}\right) - \color{blue}{\frac{0.125 \cdot 1}{{x}^{2}}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 98.7%

      \[\leadsto \left(\left(\frac{0.5}{x} + \frac{0.0625}{{x}^{3}}\right) - \frac{\color{blue}{0.125}}{{x}^{2}}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

9. Simplified 98.7%

   \[\leadsto \color{blue}{\left(\left(\frac{0.5}{x} + \frac{0.0625}{{x}^{3}}\right) - \frac{0.125}{{x}^{2}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

10. Taylor expanded in x around 0 38.2%

    \[\leadsto \color{blue}{\frac{0.0625}{{x}^{3}}} \]

11. Final simplification 38.2%

    \[\leadsto \frac{0.0625}{{x}^{3}} \]
12. Add Preprocessing

Alternative 11: 7.9% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 x))

C

double code(double x) {
	return 0.5 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 / x
end function

Java

public static double code(double x) {
	return 0.5 / x;
}

Python

def code(x):
	return 0.5 / x

Julia

function code(x)
	return Float64(0.5 / x)
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / x;
end

Wolfram

code[x_] := N[(0.5 / x), $MachinePrecision]

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 40.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   2. div-inv 40.9%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]

   3. *-un-lft-identity 40.9%

      \[\leadsto \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   4. +-commutative 40.9%

      \[\leadsto \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   5. *-rgt-identity 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   6. metadata-eval 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]

   7. frac-times 40.9%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   8. associate-*l/ 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{x + 1}}}{\sqrt{x}}} \]

   9. inv-pow 41.0%

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{1 \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   10. *-un-lft-identity 41.0%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}}{\sqrt{x}} \]

   11. sqrt-pow2 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{x}} \]

   12. +-commutative 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}}{\sqrt{x}} \]

   13. metadata-eval 40.9%

       \[\leadsto \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{\color{blue}{-0.5}}}{\sqrt{x}} \]

4. Applied egg-rr 40.9%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-*r/ 40.9%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

   2. *-rgt-identity 40.9%

      \[\leadsto \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {\left(1 + x\right)}^{-0.5}}{\color{blue}{\sqrt{x} \cdot 1}} \]

   3. times-frac 40.9%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

   4. div-sub 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} - \frac{\sqrt{x}}{\sqrt{x}}\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   5. sub-neg 40.9%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\frac{\sqrt{x}}{\sqrt{x}}\right)\right)} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   6. *-inverses 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \left(-\color{blue}{1}\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

   7. metadata-eval 40.9%

      \[\leadsto \left(\frac{\sqrt{1 + x}}{\sqrt{x}} + \color{blue}{-1}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

6. Simplified 40.9%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{x}} + -1\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1}} \]

7. Taylor expanded in x around inf 97.1%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot \frac{{\left(1 + x\right)}^{-0.5}}{1} \]

8. Taylor expanded in x around 0 7.9%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]

9. Final simplification 7.9%

   \[\leadsto \frac{0.5}{x} \]
10. Add Preprocessing

Alternative 12: 2.5% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -1.0d0
end function

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 2.5%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{1} \]

4. Taylor expanded in x around inf 2.5%

   \[\leadsto \color{blue}{-1} \]

5. Final simplification 2.5%

   \[\leadsto -1 \]
6. Add Preprocessing

Developer target: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0))))))

C

double code(double x) {
	return 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((x + 1.0d0) * sqrt(x)) + (x * sqrt((x + 1.0d0))))
end function

Java

public static double code(double x) {
	return 1.0 / (((x + 1.0) * Math.sqrt(x)) + (x * Math.sqrt((x + 1.0))));
}

Python

def code(x):
	return 1.0 / (((x + 1.0) * math.sqrt(x)) + (x * math.sqrt((x + 1.0))))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(x + 1.0) * sqrt(x)) + Float64(x * sqrt(Float64(x + 1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (((x + 1.0) * sqrt(x)) + (x * sqrt((x + 1.0))));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(x + 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024044 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))